Method and apparatus for approximating gains in dynamic and steady-state processes for prediction, control, and optimization

ABSTRACT

A method and apparatus for controlling a non-linear mill. A linear controller is provided having a linear gain k that is operable to receive inputs representing measured variables of the plant and predict on an output of the linear controller predicted control values for manipulatible variables that control the plant. A non-linear model of the plant is provided for storing a representation of the plant over a trained region of the operating input space and having a steady-state gain K associated therewith. The gain k of the linear model is adjusted with the gain K of the non-linear model in accordance with a predetermined relationship as the measured variables change the operating region of the input space at which the linear controller is predicting the values for the manipulatible variables. The predicted manipulatible variables are then output after the step of adjusting the gain k.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 11/234,927, entitled “Kiln and Thermal Combustion Control”,filed Sep. 26, 2005, which is a continuation of U.S. patent applicationSer. No.10/843,160, entitled “Kiln Thermal and Combustion Control”,filed May 11, 2004, which is a continuation of U.S. Pat. No. 6,735,483,issued May 11, 2004, Ser. No. 10/314,675, entitled “Method and Apparatusfor Controlling a Non-Linear Mill,” which is a continuation of U.S. Pat.No. 6,493,596, issued Dec. 10, 2002, Ser. No. 09/514,733, entitled“Method and Apparatus for Controlling a Non-Linear Mill,” which is acontinuation-in-part of U.S. Pat. No. 6,487,459, issued Nov. 26, 2002,Ser. No. 09/250,432, entitled “Method and Apparatus for Modeling Dynamicand Steady State Processes for Prediction, Control and Optimization,”which is a continuation of U.S. Pat. No. 5,933,345, issued Aug. 3, 1999,Ser. No. 08/643,464, entitled “Method and Apparatus for Dynamic andSteady State Modeling Over a Desired Path Between Two End Points.”

TECHNICAL FIELD OF THE INVENTION

This invention pertains in general to modeling techniques and, moreparticularly, to combining steady-state and dynamic models for thepurpose of prediction, control and optimization for non-linear millcontrol.

BACKGROUND OF THE INVENTION

Process models that are utilized for prediction, control andoptimization can be divided into two general categories, steady-statemodels and dynamic models. In each case the model is a mathematicalconstruct that characterizes the process, and process measurements areutilized to parameterize or fit the model so that it replicates thebehavior of the process. The mathematical model can then be implementedin a simulator for prediction or inverted by an optimization algorithmfor control or optimization.

Steady-state or static models are utilized in modern process controlsystems that usually store a great deal of data, this data typicallycontaining steady-state information at many different operatingconditions. The steady-state information is utilized to train anon-linear model wherein the process input variables are represented bythe vector U that is processed through the model to output the dependentvariable Y. The non-linear model is a steady-state phenomenological orempirical model developed utilizing several ordered pairs (U_(i), Y_(i))of data from different measured steady states. If a model is representedas:Y=P(U,Y)   (1)where P is some parameterization, then the steady-state modelingprocedure can be presented as:({right arrow over (U)},{right arrow over (Y)})→P   (2)where U and Y are vectors containing the U_(i), Y_(i) ordered pairelements. Given the model P, then the steady-state process gain can becalculated as: $\begin{matrix}{K = \frac{\Delta\quad{P\left( {U,Y} \right)}}{\Delta\quad U}} & (3)\end{matrix}$

The steady-state model therefore represents the process measurementsthat are taken when the system is in a “static” mode. These measurementsdo not account for the perturbations that exist when changing from onesteady-state condition to another steady-state condition. This isreferred to as the dynamic part of a model.

A dynamic model is typically a linear model and is obtained from processmeasurements which are not steady-state measurements; rather, these arethe data obtained when the process is moved from one steady-statecondition to another steady-state condition. This procedure is where aprocess input or manipulated variable u(t) is input to a process with aprocess output or controlled variable y(t) being output and measured.Again, ordered pairs of measured data (u(I), y(I)) can be utilized toparameterize a phenomenological or empirical model, this time the datacoming from non-steady-state operation. The dynamic model is representedas:y(t)=p(u(t),y(t))   (4)where p is some parameterization. Then the dynamic modeling procedurecan be represented as:({right arrow over (u)},{right arrow over (y)})→p   (5)Where u and y are vectors containing the (u(I),y(I)) ordered pairelements. Given the model p, then the steady-state gain of a dynamicmodel can be calculated as: $\begin{matrix}{k = \frac{\Delta\quad{p\left( {u,y} \right)}}{\Delta\quad u}} & (6)\end{matrix}$

Unfortunately, almost always the dynamic gain k does not equal thesteady-state gain K, since the steady-state gain is modeled on a muchlarger set of data, whereas the dynamic gain is defined around a set ofoperating conditions wherein an existing set of operating conditions aremildly perturbed. This results in a shortage of sufficient non-linearinformation in the dynamic data set in which non-linear information iscontained within the static model. Therefore, the gain of the system maynot be adequately modeled for an existing set of steady-state operatingconditions. Thus, when considering two independent models, one for thesteady-state model and one for the dynamic model, there is a mis-matchbetween the gains of the two models when used for prediction, controland optimization. The reason for this mis-match are that thesteady-state model is non-linear and the dynamic model is linear, suchthat the gain of the steady-state model changes depending on the processoperating point, with the gain of the linear model being fixed. Also,the data utilized to parameterize the dynamic model do not represent thecomplete operating range of the process, i.e., the dynamic data is onlyvalid in a narrow region. Further, the dynamic model represents theacceleration properties of the process (like inertia) whereas thesteady-state model represents the tradeoffs that determine the processfinal resting value (similar to the tradeoff between gravity and dragthat determines terminal velocity in free fall).

One technique for combining non-linear static models and linear dynamicmodels is referred to as the Hammerstein model. The Hammerstein model isbasically an input-output representation that is decomposed into twocoupled parts. This utilizes a set of intermediate variables that aredetermined by the static models which are then utilized to construct thedynamic model. These two models are not independent and are relativelycomplex to create.

SUMMARY OF THE INVENTION

The present invention disclosed and claimed herein comprises a methodfor controlling a non-linear plant. A linear controller is providedhaving a linear gain k that is operable to receive inputs representingmeasured variables of the plant and predict on an output of the linearcontroller predicted control values for manipulatible variables thatcontrol the plant. A non-linear model of the plant is provided forstoring a representation of the plant over a trained region of theoperating input space and having a steady-state gain K associatedtherewith. The gain k of the linear model is adjusted with the gain K ofthe non-linear model in accordance with a predetermined relationship asthe measured variables change the operating region of the input space atwhich the linear controller is predicting the values for themanipulatible variables. The predicted manipulatible variables are thenoutput after the step of adjusting the gain k.

DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention and theadvantages thereof, reference is now made to the following descriptiontaken in conjunction with the accompanying Drawings in which:

FIG. 1 illustrates a prior art Hammerstein model;

FIG. 2 illustrates a block diagram of the modeling technique of thepresent invention;

FIG. 3 a-3 d illustrate timing diagrams for the various outputs of thesystem of FIG. 2;

FIG. 4 illustrates a detailed block diagram of the dynamic modelutilizing the identification method;

FIG. 5 illustrates a block diagram of the operation of the model of FIG.4;

FIG. 6 illustrates an example of the modeling technique of the presentinvention utilized in a control environment;

FIG. 7 illustrates a diagrammatic view of a change between twosteady-state values;

FIG. 8 illustrates a diagrammatic view of the approximation algorithmfor changes in the steady-state value;

FIG. 9 illustrates a block diagram of the dynamic model;

FIG. 10 illustrates a detail of the control network utilizing the errorconstraining algorithm of the present invention;

FIGS. 11 a and 11 b illustrate plots of the input and output duringoptimization;

FIG. 12 illustrates a plot depicting desired and predicted behavior;

FIG. 13 illustrates various plots for controlling a system to force thepredicted behavior to the desired behavior;

FIG. 14 illustrates a plot of the trajectory weighting algorithm of thepresent invention;

FIG. 15 illustrates a plot for the constraining algorithm;

FIG. 16 illustrates a plot of the error algorithm as a function of time;

FIG. 17 illustrates a flowchart depicting the statistical method forgenerating the filter and defining the end point for the constrainingalgorithm of FIG. 15;

FIG. 18 illustrates a diagrammatic view of the optimization process;

FIG. 18 a illustrates a diagrammatic representation of the manner inwhich the path between steady-state values is mapped through the inputand output space;

FIG. 19 illustrates a flowchart for the optimization procedure;

FIG. 20 illustrates a diagrammatic view of the input space and the errorassociated therewith;

FIG. 21 illustrates a diagrammatic view of the confidence factor in theinput space;

FIG. 22 illustrates a block diagram of the method for utilizing acombination of a non-linear system and a first principal system; and

FIG. 23 illustrates an alternate embodiment of the embodiment of FIG.22.

FIG. 24 illustrates a block diagram of a kiln, cooler and preheater;

FIG. 25 illustrates a block diagram of a non-linear controlled mill;

FIG. 26 illustrates a table for various aspects of the mill associatedwith the fresh feed and the separator speed;

FIG. 27 illustrates historical modeling data for the mill;

FIGS. 28 a and 28 b illustrate diagrams of sensitivity versus thepercent of the Blaine and the percent of the Return, respectively;

FIG. 29 illustrates non-linear gains for fresh feed responses;

FIG. 30 illustrates plots of non-linear gains for separator speedresponses; and

FIG. 31 illustrates a closed loop non-linear model predictive control,target change for Blaine.

DETAILED DESCRIPTION OF THE INVENTION

Referring now to FIG. 1, there is illustrated a diagrammatic view of aHammerstein model of the prior art. This is comprised of a non-linearstatic operator model 10 and a linear dynamic model 12, both disposed ina series configuration. The operation of this model is described in H.T. Su, and T. J. McAvoy, “Integration of Multilayer Perceptron Networksand Linear Dynamic Models: A Hammerstein Modeling Approach” to appear inI & EC Fundamentals, paper dated Jul. 7, 1992, which reference isincorporated herein by reference. Hammerstein models in general havebeen utilized in modeling non-linear systems for some time. Thestructure of the Hammerstein model illustrated in FIG. 1 utilizes thenon-linear static operator model 10 to transform the input U intointermediate variables H. The non-linear operator is usually representedby a finite polynomial expansion. However, this could utilize a neuralnetwork or any type of compatible modeling system. The linear dynamicoperator model 12 could utilize a discreet dynamic transfer functionrepresenting the dynamic relationship between the intermediate variableH and the output Y. For multiple input systems, the non-linear operatorcould utilize a multilayer neural network, whereas the linear operatorcould utilize a two layer neural network. A neural network for thestatic operator is generally well known and described in U.S. Pat. No.5,353,207, issued Oct. 4, 1994, and assigned to the present assignee,which is incorporated herein by reference. These type of networks aretypically referred to as a multilayer feed-forward network whichutilizes training in the form of back-propagation. This is typicallyperformed on a large set of training data. Once trained the network hasweights associated therewith, which are stored in a separate database.

Once the steady-state model is obtained, one can then choose the outputvector from the hidden layer in the neural network as the intermediatevariable for the Hammerstein model. In order to determine the input forthe linear dynamic operator, u(t), it is necessary to scale the outputvector h(d) from the non-linear static operator model 10 for the mappingof the intermediate variable h(t) to the output variable of the dynamicmodel y(t); which is determined by the linear dynamic model.

During the development of a linear dynamic model to represent the lineardynamic operator, in the Hammerstein model, it is important that thesteady-state non-linearity remain the same. To achieve this goal, onemust train the dynamic model subject to a constraint so that thenon-linearity learned by the steady-state model remains unchanged afterthe training. This results in a dependency of the two models on eachother.

Referring now to FIG. 2, there is illustrated a block diagram of themodeling method of the present invention, which is referred to as asystematic modeling technique. The general concept of the systematicmodeling technique in the present invention results from the observationthat, while process gains (steady-state behavior) vary with U's and Y's,( i.e., the gains are non-linear), the process dynamics seemingly varywith time only, (i.e., they can be modeled as locally linear, buttime-varied). By utilizing non-linear models for the steady-statebehavior and linear models for the dynamic behavior, several practicaladvantages result. They are as follows:

1. Completely rigorous models can be utilized for the steady-state part.This provides a credible basis for economic optimization.

2. The linear models for the dynamic part can be updated on-line, i.e.,the dynamic parameters that are known to be time-varying can be adaptedslowly.

3. The gains of the dynamic models and the gains of the steady-statemodels can be forced to be consistent (k=K).

With further reference to FIG. 2, there are provided a static orsteady-state model 20 and a dynamic model 22. The static model 20, asdescribed above, is a rigorous model that is trained on a large set ofsteady-state data. The static model 20 will receive a process input Uand provide a predicted output Y. These are essentially steady-statevalues. The steady-state values at a given time are latched in variouslatches, an input latch 24 and an output latch 26. The latch 24 containsthe steady-state value of the input U_(ss), and the latch 26 containsthe steady-state output value Y_(ss). The dynamic model 22 is utilizedto predict the behavior of the plant when a change is made from asteady-state value of Y_(ss) to a new value Y. The dynamic model 22receives on the input the dynamic input value u and outputs a predicteddynamic value y. The value u is comprised of the difference between thenew value U and the steady-state value in the latch 24, U_(ss). This isderived from a subtraction circuit 30 which receives on the positiveinput thereof the output of the latch 24 and on the negative inputthereof the new value of U. This therefore represents the delta changefrom the steady-state. Similarly, on the output the predicted overalldynamic value will be the sum of the output value of the dynamic model,y, and the steady-state output value stored in the latch 26, Y_(ss).These two values are summed with a summing block 34 to provide apredicted output Y. The difference between the value output by thesumming junction 34 and the predicted value output by the static model20 is that the predicted value output by the summing junction 20accounts for the dynamic operation of the system during a change. Forexample, to process the input values that are in the input vector U bythe static model 20, the rigorous model, can take significantly moretime than running a relatively simple dynamic model. The method utilizedin the present invention is to force the gain of the dynamic model 22k_(d) to equal the gain K_(ss) of the static model 20.

In the static model 20, there is provided a storage block 36 whichcontains the static coefficients associated with the static model 20 andalso the associated gain value K_(ss). Similarly, the dynamic model 22has a storage area 38 that is operable to contain the dynamiccoefficients and the gain value k_(d). One of the important aspects ofthe present invention is a link block 40 that is operable to modify thecoefficients in the storage area 38 to force the value of k_(d) to beequal to the value of k_(ss). Additionally, there is an approximationblock 41 that allows approximation of the dynamic gain k_(d) between themodification updates.

Systematic Model

The linear dynamic model 22 can generally be represented by thefollowing equations: $\begin{matrix}{{\delta\quad{y(t)}} = {{\sum\limits_{i = 1}^{n}{b_{i}\delta\quad{u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}{a_{i}\delta\quad{y\left( {t - i} \right)}}}}} & (7)\end{matrix}$where:δy(t)=y(t)−Y _(ss)   (8)δu(t)=u(t)−U _(ss)   (9)and t is time, a_(i) and b_(i) are real numbers, d is a time delay, u(t)is an input and y(t) an output. The gain is represented by:$\begin{matrix}{\frac{y(B)}{u(B)} = {k = \frac{\left( {\sum\limits_{i = 1}^{n}{b_{i}B^{i - 1}}} \right)B^{d}}{1 + {\sum\limits_{i = 1}^{n}{a_{i}B^{i - 1}}}}}} & (10)\end{matrix}$where B is the backward shift operator B(x(t))=x(t−1), t=time, the a_(i)and b_(i) are real numbers, I is the number of discreet time intervalsin the dead-time of the process, and n is the order of the model. Thisis a general representation of a linear dynamic model, as contained inGeorge E. P. Box and G. M Jenkins, “TIME SERIES ANALYSIS forecasting andcontrol”, Holden-Day, San Francisco, 1976, Section 10.2, Page 345. Thisreference is incorporated herein by reference.

The gain of this model can be calculated by setting the value of B equalto a value of “1”. The gain will then be defined by the followingequation: $\begin{matrix}{\left\lbrack \frac{y(B)}{u(B)} \right\rbrack_{B = 1} = {k_{d} = \frac{\sum\limits_{i = 1}^{n}b_{i}}{1 + {\sum\limits_{i = 1}^{n}a_{i}}}}} & (11)\end{matrix}$

The a_(i) contain the dynamic signature of the process, its unforced,natural response characteristic. They are independent of the processgain. The b_(i) contain part of the dynamic signature of the process;however, they alone contain the result of the forced response. The b_(i)determine the gain k of the dynamic model. See: J. L. Shearer, A. T.Murphy, and H. H. Richardson, “Introduction to System Dynamics”,Addison-Wesley, Reading, Mass., 1967, Chapter 12. This reference isincorporated herein by reference.

Since the gain K_(ss) of the steady-state model is known, the gain k_(d)of the dynamic model can be forced to match the gain of the steady-statemodel by scaling the b_(i) parameters. The values of the static anddynamic gains are set equal with the value of b_(i) scaled by the ratioof the two gains: $\begin{matrix}{\left( b_{i} \right)_{scaled} = {\left( b_{i} \right)_{old}\left( \frac{K_{ss}}{k_{d}} \right)}} & (12)\end{matrix}$ $\begin{matrix}{\left( b_{i} \right)_{scaled} = \frac{\left( b_{i} \right)_{old}{K_{ss}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}}} & (13)\end{matrix}$

This makes the dynamic model consistent with its steady-statecounterpart. Therefore, each time the steady-state value changes, thiscorresponds to a gain K_(ss) of the steady-state model. This value canthen be utilized to update the gain k_(d) of the dynamic model and,therefore, compensate for the errors associated with the dynamic modelwherein the value of k_(d) is determined based on perturbations in theplant on a given set of operating conditions. Since all operatingconditions are not modeled, the step of varying the gain will accountfor changes in the steady-state starting points.

Referring now to FIGS. 3 a-3 d, there are illustrated plots of thesystem operating in response to a step function wherein the input valueU changes from a value of 100 to a value of 110. In FIG. 3 a, the valueof 100 is referred to as the previous steady-state value U_(ss). In FIG.3 b, the value of u varies from a value of 0 to a value of 10, thisrepresenting the delta between the steady-state value of U_(ss) to thelevel of 110, represented by reference numeral 42 in FIG. 3 a.Therefore, in FIG. 3 b the value of u will go from 0 at a level 44, to avalue of 10 at a level 46. In FIG. 3 c, the output Y is represented ashaving a steady-state value Y_(ss) of 4 at a level 48. When the inputvalue U rises to the level 42 with a value of 110, the output value willrise. This is a predicted value. The predicted value which is the properoutput value is represented by a level 50, which level 50 is at a valueof 5. Since the steady-state value is at a value of 4, this means thatthe dynamic system must predict a difference of a value of 1. This isrepresented by FIG. 3 d wherein the dynamic output value y varies from alevel 54 having a value of 0 to a level 56 having a value of 1.0.However, without the gain scaling, the dynamic model could, by way ofexample, predict a value for y of 1.5, represented by dashed level 58,if the steady-state values were outside of the range in which thedynamic model was trained. This would correspond to a value of 5.5 at alevel 60 in the plot of FIG. 3 c. It can be seen that the dynamic modelmerely predicts the behavior of the plant from a starting point to astopping point, not taking into consideration the steady-state values.It assumes that the steady-state values are those that it was trainedupon. If the gain k_(d) were not scaled, then the dynamic model wouldassume that the steady-state values at the starting point were the samethat it was trained upon. However, the gain scaling link between thesteady-state model and the dynamic model allow the gain to be scaled andthe parameter b_(i) to be scaled such that the dynamic operation isscaled and a more accurate prediction is made which accounts for thedynamic properties of the system.

Referring now to FIG. 4, there is illustrated a block diagram of amethod for determining the parameters a_(i), b_(i). This is usuallyachieved through the use of an identification algorithm, which isconventional. This utilizes the (u(t),y(t)) pairs to obtain the a_(i)and b_(i) parameters. In the preferred embodiment, a recursiveidentification method is utilized where the a_(i) and b_(i) parametersare updated with each new (u_(i)(t),y_(i)(t)) pair. See: T. Eykhoff,“System Identification”, John-Wiley & Sons, New York, 1974, Pages 38 and39, et. seq., and H. Kurz and W. Godecke, “Digital Parameter-AdaptiveControl Processes with Unknown Dead Time”, Automatica, Vol. 17, No. 1,1981, pp. 245-252, which references are incorporated herein byreference.

In the technique of FIG. 4, the dynamic model 22 has the output thereofinput to a parameter-adaptive control algorithm block 60 which adjuststhe parameters in the coefficient storage block 38, which also receivesthe scaled values of k, b_(i). This is a system that is updated on aperiodic basis, as defined by timing block 62. The control algorithm 60utilizes both the input u and the output y for the purpose ofdetermining and updating the parameters in the storage area 38.

Referring now to FIG. 5, there is illustrated a block diagram of thepreferred method. The program is initiated in a block 68 and thenproceeds to a function block 70 to update the parameters a_(i), b_(i)utilizing the (u(I),y(I)) pairs. Once these are updated, the programflows to a function block 72 wherein the steady-state gain factor K isreceived, and then to a function block 74 to set the dynamic gain to thesteady state gain, i.e., provide the scaling function describedhereinabove. This is performed after the update. This procedure can beused for on-line identification, non-linear dynamic model prediction andadaptive control.

Referring now to FIG. 6, there is illustrated a block diagram of oneapplication of the present invention utilizing a control environment. Aplant 78 is provided which receives input values u(t) and outputs anoutput vector y(t). The plant 78 also has measurable state variabless(t). A predictive model 80 is provided which receives the input valuesu(t) and the state variables s(t) in addition to the output value y(t).The steady-state model 80 is operable to output a predicted value ofboth y(t) and also of a future input value u(t+1). This constitutes asteady-state portion of the system. The predicted steady-state inputvalue is U_(ss) with the predicted steady-state output value beingY_(ss). In a conventional control scenario, the steady-state model 80would receive as an external input a desired value of the outputy^(d)(t) which is the desired value that the overall control systemseeks to achieve. This is achieved by controlling a distributed controlsystem (DCS) 86 to produce a desired input to the plant. This isreferred to as u(t+1), a future value. Without considering the dynamicresponse, the predictive model 80, a steady-state model, will providethe steady-state values. However, when a change is desired, this changewill effectively be viewed as a “step response”.

To facilitate the dynamic control aspect, a dynamic controller 82 isprovided which is operable to receive the input u(t), the output valuey(t) and also the steady-state values U_(ss) and Y_(ss) and generate theoutput u(t+1). The dynamic controller effectively generates the dynamicresponse between the changes, i.e., when the steady-state value changesfrom an initial steady-state value U_(ss) ^(i), Y^(i) _(ss) to a finalsteady-state value U^(f) _(ss), Y^(f) _(ss).

During the operation of the system, the dynamic controller 82 isoperable in accordance with the embodiment of FIG. 2 to update thedynamic parameters of the dynamic controller 82 in a block 88 with again link block 90, which utilizes the value K_(ss) from a steady-stateparameter block in order to scale the parameters utilized by the dynamiccontroller 82, again in accordance with the above described method. Inthis manner, the control function can be realized. In addition, thedynamic controller 82 has the operation thereof optimized such that thepath traveled between the initial and final steady-state values isachieved with the use of the optimizer 83 in view of optimizerconstraints in a block 85. In general, the predicted model (steady-statemodel) 80 provides a control network function that is operable topredict the future input values. Without the dynamic controller 82, thisis a conventional control network which is generally described in U.S.Pat. No. 5,353,207, issued Oct. 4, 1994, to the present assignee, whichpatent is incorporated herein by reference.

Approximate Systematic Modeling

For the modeling techniques described thus far, consistency between thesteady-state and dynamic models is maintained by resealing the b_(i)parameters at each time step utilizing equation 13. If the systematicmodel is to be utilized in a Model Predictive Control (MPC) algorithm,maintaining consistency may be computationally expensive. These types ofalgorithms are described in C. E. Garcia, D. M. Prett and M. Morari.Model predictive control: theory and practice—a survey, Automatica,25:335-348, 1989; D. E. Seborg, T. F. Edgar, and D. A. Mellichamp.Process Dynamics and Control. John Wiley and Sons, New York, N.Y., 1989.These references are incorporated herein by reference. For example, ifthe dynamic gain k_(d) is computed from a neural network steady-statemodel, it would be necessary to execute the neural network module eachtime the model was iterated in the MPC algorithm. Due to the potentiallylarge number of model iterations for certain MPC problems, it could becomputationally expensive to maintain a consistent model. In this case,it would be better to use an approximate model which does not rely onenforcing consistencies at each iteration of the model.

Referring now to FIG. 7, there is illustrated a diagram for a changebetween steady state values. As illustrated, the steady-state model willmake a change from a steady-state value at a line 100 to a steady-statevalue at a line 102. A transition between the two steady-state valuescan result in unknown settings. The only way to insure that the settingsfor the dynamic model between the two steady-state values, an initialsteady-state value K_(ss) ^(i) and a final steady-state gain K_(ss)^(f), would be to utilize a step operation, wherein the dynamic gaink_(d) was adjusted at multiple positions during the change. However,this may be computationally expensive. As will be described hereinbelow,an approximation algorithm is utilized for approximating the dynamicbehavior between the two steady-state values utilizing a quadraticrelationship. This is defined as a behavior line 104, which is disposedbetween an envelope 106, which behavior line 104 will be describedhereinbelow.

Referring now to FIG. 8, there is illustrated a diagrammatic view of thesystem undergoing numerous changes in steady-state value as representedby a stepped line 108. The stepped line 108 is seen to vary from a firststeady-state value at a level 110 to a value at a level 112 and thendown to a value at a level 114, up to a value at a level 116 and thendown to a final value at a level 118. Each of these transitions canresult in unknown states. With the approximation algorithm that will bedescribed hereinbelow, it can be seen that, when a transition is madefrom level 110 to level 112, an approximation curve for the dynamicbehavior 120 is provided. When making a transition from level 114 tolevel 116, an approximation gain curve 124 is provided to approximatethe steady state gains between the two levels 114 and 116. For makingthe transition from level 116 to level 118, an approximation gain curve126 for the steady-state gain is provided. It can therefore be seen thatthe approximation curves 120-126 account for transitions betweensteady-state values that are determined by the network, it being notedthat these are approximations which primarily maintain the steady-stategain within some type of error envelope, the envelope 106 in FIG. 7.

The approximation is provided by the block 41 noted in FIG. 2 and can bedesigned upon a number of criteria, depending upon the problem that itwill be utilized to solve. The system in the preferred embodiment, whichis only one example, is designed to satisfy the following criteria:

1. Computational Complexity: The approximate systematic model will beused in a Model Predictive Control algorithm, therefore, it is requiredto have low computational complexity.

2. Localized Accuracy: The steady-state model is accurate in localizedregions. These regions represent the steady-state operating regimes ofthe process. The steady-state model is significantly less accurateoutside these localized regions.

3. Final Steady-State: Given a steady-state set point change, anoptimization algorithm which uses the steady-state model will be used tocompute the steady-state inputs required to achieve the set point.

Because of item 2, it is assumed that the initial and finalsteady-states associated with a set-point change are located in regionsaccurately modeled by the steady-state model.

Given the noted criteria, an approximate systematic model can beconstructed by enforcing consistency of the steady-state and dynamicmodel at the initial and final steady-state associated with a set pointchange and utilizing a linear approximation at points in between the twosteady-states. This approximation guarantees that the model is accuratein regions where the steady-state model is well known and utilizes alinear approximation in regions where the steady-state model is known tobe less accurate. In addition, the resulting model has low computationalcomplexity. For purposes of this proof, Equation 13 is modified asfollows: $\begin{matrix}{b_{i,{scaled}} = \frac{b_{i}{K_{ss}\left( {u\left( {t - d - 1} \right)} \right)}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}{\sum\limits_{i = 1}^{n}b_{i}}} & (14)\end{matrix}$

This new equation 14 utilizes K_(ss)(u(t−d−1)) instead of K_(ss)(u(t))as the consistent gain, resulting in a systematic model which is delayinvariant.

The approximate systematic model is based upon utilizing the gainsassociated with the initial and final steady-state values of a set-pointchange. The initial steady-state gain is denoted K^(i) _(ss) while theinitial steady-state input is given by U^(i) _(ss). The finalsteady-state gain is K^(f) _(ss) and the final input is U^(f) _(ss).Given these values, a linear approximation to the gain is given by:$\begin{matrix}{{K_{ss}\left( {u(t)} \right)} = {K_{ss}^{i} + {\frac{K_{ss}^{f} - K_{ss}^{i}}{U_{ss}^{f} - U_{ss}^{i}}{\left( {{u(t)} - U_{ss}^{i}} \right).}}}} & (15)\end{matrix}$

Substituting this approximation into Equation 13 and replacingu(t−d−1)−u^(i) by δu(t−d−1) yields: $\begin{matrix}{{\overset{\sim}{b}}_{j,{scaled}} = {\frac{b_{j}{K_{ss}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}} + {\frac{1}{2}\frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}\delta\quad{{u\left( {t - d - i} \right)}.}}}} & (16)\end{matrix}$

To simplify the expression, define the variable b_(j)-Bar as:$\begin{matrix}{\overset{\_}{b_{j}} = \frac{b_{j}{K_{ss}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}}} & (17)\end{matrix}$and g_(j) as: $\begin{matrix}{g_{j} = \frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}} & (18)\end{matrix}$

Equation 16 may be written as:{tilde over (b)} _(j,scaled) ={tilde over (b)} _(j) +g _(j) δu(t−d−i).  (19)

Finally, substituting the scaled b's back into the original differenceEquation 7, the following expression for the approximate systematicmodel is obtained: $\begin{matrix}{{\delta\quad{y(t)}} = {{\sum\limits_{i = 1}^{n}{\overset{\_}{b_{i}}\delta\quad{u\left( {t - d - i} \right)}}} + {\sum\limits_{i = 1}^{n}{g_{i}\delta\quad{u\left( {t - d - i^{2}} \right)}\delta\quad{u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}{a_{i}\delta\quad{y\left( {t - i} \right)}}}}} & (20)\end{matrix}$

The linear approximation for gain results in a quadratic differenceequation for the output. Given Equation 20, the approximate systematicmodel is shown to be of low computational complexity. It may be used ina MPC algorithm to efficiently compute the required control moves for atransition from one steady-state to another after a set-point change.Note that this applies to the dynamic gain variations betweensteady-state transitions and not to the actual path values.

Control System Error Constraints

Referring now to FIG. 9, there is illustrated a block diagram of thepredictin engine for the dynamic controller 82 of FIG. 6. The predictionengine is operable to essentially predict a value of y(t) as thepredicted future value y(t+1). Since the prediction engine mustdetermine what the value of the output y(t) is at each future valuebetween two steady-state values, it is necessary to perform these in a“step” manner. Therefore, there will be k steps from a value of zero toa value of N, which value at k=N is the value at the “horizon”, thedesired value. This, as will be described hereinbelow, is an iterativeprocess, it being noted that the terminology for “(t+1)” refers to anincremental step, with an incremental step for the dynamic controllerbeing smaller than an incremented step for the steady-state model. Forthe steady-state model, “y(t+N)” for the dynamic model will be, “y(t+1)”for the steady state The value y(t+1) is defined as follows:y(t+1)=a ₁ y(t)+a ₂ y(t−1)+b ₁ u(t−d−1)+b ₂ u(t−d−2)   (21)

With further reference to FIG. 9, the input values u(t) for each (u,y)pair are input to a delay line 140. The output of the delay lineprovides the input value u(t) delayed by a delay value “d”. There areprovided only two operations for multiplication with the coefficients b₁and b₂, such that only two values u(t) and u(t−1) are required. Theseare both delayed and then multiplied by the coefficients b₁ and b₂ andthen input to a summing block 141. Similarly, the output value y^(p)(t)is input to a delay line 142, there being two values required formultiplication with the coefficients a₁ and a₂. The output of thismultiplication is then input to the summing block 141. The input to thedelay line 142 is either the actual input value y^(a)(t) or the iteratedoutput value of the summation block 141, which is the previous valuecomputed by the dynamic controller 82. Therefore, the summing block 141will output the predicted value y(t+1) which will then be input to amultiplexor 144. The multiplexor 144 is operable to select the actualoutput y^(a)(t) on the first operation and, thereafter, select theoutput of the summing block 141. Therefore, for a step value of k=0 thevalue y^(a)(t) will be selected by the multiplexor 144 and will belatched in a latch 145. The latch 145 will provide the predicted valuey^(p)(t+k) on an output 146. This is the predicted value of y(t) for agiven k that is input back to the input of delay line 142 formultiplication with the coefficients a₁ and a₂. This is iterated foreach value of k from k=0 to k=N.

The a₁ and a₂ values are fixed, as described above, with the b₁ and b₂values scaled. This scaling operation is performed by the coefficientmodification block 38. However, this only defines the beginningsteady-state value and the final steady-state value, with the dynamiccontroller and the optimization routines described in the presentapplication defining how the dynamic controller operates between thesteady-state values and also what the gain of the dynamic controller is.The gain specifically is what determines the modification operationperformed by the coefficient modification block 38.

In FIG. 9, the coefficients in the coefficient modification block 38 aremodified as described hereinabove with the information that is derivedfrom the steady-state model. The steady-state model is operated in acontrol application, and is comprised in part of a forward steady-statemodel 141 which is operable to receive the steady-state input valueU_(ss)(t) and predict the steady-state output value Y_(ss)(t). Thispredicted value is utilized in an inverse steady-state model 143 toreceive the desired value y^(d)(t) and the predicted output of thesteady-state model 141 and predict a future steady-state input value ormanipulated value U_(ss)(t+N) and also a future steady-state input valueY_(ss)(t+N) in addition to providing the steady-state gain K_(ss). Asdescribed hereinabove, these are utilized to generate scaled b-values.These b-values are utilized to define the gain k_(d) of the dynamicmodel. In can therefore be seen that this essentially takes a lineardynamic model with a fixed gain and allows it to have a gain thereofmodified by a non-linear model as the operating point is moved throughthe output space.

Referring now to FIG. 10, there is illustrated a block diagram of thedynamic controller and optimizer. The dynamic controller includes adynamic model 149 which basically defines the predicted value y^(p)(k)as a function of the inputs y(t), s(t) and u(t). This was essentiallythe same model that was described hereinabove with reference to FIG. 9.The model 149 predicts the output values y^(p)(k) between the twosteady-state values, as will be described hereinbelow. The model 149 ispredefined and utilizes an identification algorithm to identify the a₁,a₂, b₁ and b₂ coefficients during training. Once these are identified ina training and identification procedure, these are “fixed”. However, asdescribed hereinabove, the gain of the dynamic model is modified byscaling the coefficients b₁ and b₂. This gain scaling is not describedwith respect to the optimization operation of FIG. 10, although it canbe incorporated in the optimization operation.

The output of model 149 is input to the negative input of a summingblock 150. Summing block 150 sums the predicted output y^(p)(k) with thedesired output y^(d)(t). In effect, the desired value of y^(d)(t) iseffectively the desired steady-state value Y^(f) _(ss), although it canbe any desired value. The output of the summing block 150 comprises anerror value which is essentially the difference between the desiredvalue y^(d)(t) and the predicted value y^(p)(k). The error value ismodified by an error modification block 151, as will be describedhereinbelow, in accordance with error modification parameters in a block152. The modified error value is then input to an inverse model 153,which basically performs an optimization routine to predict a change inthe input value u(t). In effect, the optimizer 153 is utilized inconjunction with the model 149 to minimize the error output by summingblock 150. Any optimization function can be utilized, such as a MonteCarlo procedure. However, in the present invention, a gradientcalculation is utilized. In the gradient method, the gradient ∂(y)/∂(u)is calculated and then a gradient solution performed as follows:$\begin{matrix}{{\Delta\quad u_{new}} = {{\Delta\quad u_{old}} + {\left( \frac{\partial(y)}{\partial(u)} \right) \times E}}} & (22)\end{matrix}$

The optimization function is performed by the inverse model 153 inaccordance with optimization constraints in a block 154. An iterationprocedure is performed with an iterate block 155 which is operable toperform an iteration with the combination of the inverse model 153 andthe predictive model 149 and output on an output line 156 the futurevalue u(t+k+1). For k=0, this will be the initial steady-state value andfor k=N, this will be the value at the horizon, or at the nextsteady-state value. During the iteration procedure, the previous valueof u(t+k) has the change value Δu added thereto. This value is utilizedfor that value of k until the error is within the appropriate levels.Once it is at the appropriate level, the next u(t+k) is input to themodel 149 and the value thereof optimized with the iterate block 155.Once the iteration procedure is done, it is latched. As will bedescribed hereinbelow, this is a combination of modifying the error suchthat the actual error output by the block 150 is not utilized by theoptimizer 153 but, rather, a modified error is utilized. Alternatively,different optimization constraints can be utilized, which are generatedby the block 154, these being described hereinbelow.

Referring now to FIGS. 11 a and 11 b, there are illustrated plots of theoutput y(t+k) and the input u_(k)(t+k+1), for each k from the initialsteady-state value to the horizon steady-state value at k=N. Withspecific reference to FIG. 11 a, it can be seen that the optimizationprocedure is performed utilizing multiple passes. In the first pass, theactual value u^(a)(t+k) for each k is utilized to determine the valuesof y(t+k) for each u,y pair. This is then accumulated and the valuesprocessed through the inverse model 153 and the iterate block 155 tominimize the error. This generates a new set of inputs u_(k)(t+k+1)illustrated in FIG. 11 b. Therefore, the optimization after pass 1generates the values of u(t+k+1) for the second pass. In the secondpass, the values are again optimized in accordance with the variousconstraints to again generate another set of values for u(t+k+1). Thiscontinues until the overall objective function is reached. Thisobjective function is a combination of the operations as a function ofthe error and the operations as a function of the constraints, whereinthe optimization constraints may control the overall operation of theinverse model 153 or the error modification parameters in block 152 maycontrol the overall operation. Each of the optimization constraints willbe described in more detail hereinbelow.

Referring now to FIG. 12, there is illustrated a plot of y^(d)(t) andy^(p)(t). The predicted value is represented by a waveform 170 and thedesired output is represented by a waveform 172, both plotted over thehorizon between an initial steady-state value Y^(i) _(ss) and a finalsteady-state value Y^(f) _(ss). It can be seen that the desired waveformprior to k=0 is substantially equal to the predicted output. At k=0, thedesired output waveform 172 raises its level, thus creating an error. Itcan be seen that at k=0, the error is large and the system then mustadjust the manipulated variables to minimize the error and force thepredicted value to the desired value. The objective function for thecalculation of error is of the form: $\begin{matrix}{\min\limits_{\Delta\quad u_{il}}{\sum\limits_{j}{\sum\limits_{k}\left( {A_{j}*\left( {{{\overset{\rightarrow}{y}}^{p}(t)} - {{\overset{\rightarrow}{y}}^{d}(t)}} \right)^{2}} \right.}}} & (23)\end{matrix}$where: Du_(il) is the change in input variable (IV) I at time interval l

A_(j) is the weight factor for control variable (CV) j

y^(p)(t) is the predicted value of CV j at time interval k

y^(d)(t) is the desired value of CV j.

Trajectory Weighting

The present system utilizes what is referred to as “trajectoryweighting” which encompasses the concept that one does not put aconstant degree of importance on the future predicted process behaviormatching the desired behavior at every future time set, i.e., at lowk-values. One approach could be that one is more tolerant of error inthe near term (low k-values) than farther into the future (highk-values). The basis for this logic is that the final desired behavioris more important than the path taken to arrive at the desired behavior,otherwise the path traversed would be a step function. This isillustrated in FIG. 13 wherein three possible predicted behaviors areillustrated, one represented by a curve 174 which is acceptable, one isrepresented by a different curve 176, which is also acceptable and onerepresented by a curve 178, which is unacceptable since it goes abovethe desired level on curve 172. Curves 174-178 define the desiredbehavior over the horizon for k=1 to N.

In Equation 23, the predicted curves 174-178 would be achieved byforcing the weighting factors A_(j) to be time varying. This isillustrated in FIG. 14. In FIG. 14, the weighting factor A as a functionof time is shown to have an increasing value as time and the value of kincreases. This results in the errors at the beginning of the horizon(low k-values) being weighted much less than the errors at the end ofthe horizon (high k-values). The result is more significant than merelyredistributing the weights out to the end of the control horizon at k=N.This method also adds robustness, or the ability to handle a mismatchbetween the process and the prediction model. Since the largest error isusually experienced at the beginning of the horizon, the largest changesin the independent variables will also occur at this point. If there isa mismatch between the process and the prediction (model error), theseinitial moves will be large and somewhat incorrect, which can cause poorperformance and eventually instability. By utilizing the trajectoryweighting method, the errors at the beginning of the horizon areweighted less, resulting in smaller changes in the independent variablesand, thus, more robustness.

Error Constraints

Referring now to FIG. 15, there are illustrated constraints that can beplaced upon the error. There is illustrated a predicted curve 180 and adesired curve 182, desired curve 182 essentially being a flat line. Itis desirable for the error between curve 180 and 182 to be minimized.Whenever a transient occurs at t=0, changes of some sort will berequired. It can be seen that prior to t=0, curve 182 and 180 aresubstantially the same, there being very little error between the two.However, after some type of transition, the error will increase. If arigid solution were utilized, the system would immediately respond tothis large error and attempt to reduce it in as short a time aspossible. However, a constraint frustum boundary 184 is provided whichallows the error to be large at t=0 and reduces it to a minimum level ata point 186. At point 186, this is the minimum error, which can be setto zero or to a non-zero value, corresponding to the noise level of theoutput variable to be controlled. This therefore encompasses the sameconcepts as the trajectory weighting method in that final futurebehavior is considered more important that near term behavior. The evershrinking minimum and/or maximum bounds converge from a slack positionat t=0 to the actual final desired behavior at a point 186 in theconstraint frustum method.

The difference between constraint frustums and trajectory weighting isthat constraint frustums are an absolute limit (hard constraint) whereany behavior satisfying the limit is just as acceptable as any otherbehavior that also satisfies the limit. Trajectory weighting is a methodwhere differing behaviors have graduated importance in time. It can beseen that the constraints provided by the technique of FIG. 15 requiresthat the value y^(p)(t) is prevented from exceeding the constraintvalue. Therefore, if the difference between y^(d)(t) and y^(p)(t) isgreater than that defined by the constraint boundary, then theoptimization routine will force the input values to a value that willresult in the error being less than the constraint value. In effect,this is a “clamp” on the difference between y^(p)(t) and y^(d)(t). Inthe trajectory weighting method, there is no “clamp” on the differencetherebetween; rather, there is merely an attenuation factor placed onthe error before input to the optimization network.

Trajectory weighting can be compared with other methods, there being twomethods that will be described herein, the dynamic matrix control (DMC)algorithm and the identification and command (IdCom) algorithm. The DMCalgorithm utilizes an optimization to solve the control problem byminimizing the objective function: $\begin{matrix}{\min\limits_{\Delta\quad U_{il}}{\sum\limits_{j}{\sum\limits_{k}\left( {{A_{j}*\left( {{{\overset{\rightarrow}{y}}^{P}(t)} - {{\overset{\rightarrow}{y}}^{D}(t)}} \right)} + {\sum\limits_{i}{B_{i}*{\sum\limits_{1}\left( {\Delta\quad U_{il}} \right)^{2}}}}} \right.}}} & (24)\end{matrix}$where B_(i) is the move suppression factor for input variable I. This isdescribed in Cutler, C. R. and B. L. Ramaker, Dynamic Matrix Control—AComputer Control Algorithm, AIChE National Meeting, Houston, Tex.(April, 1979), which is incorporated herein by reference.

It is noted that the weights A_(j) and desired values y^(d)(t) areconstant for each of the control variables. As can be seen from Equation24, the optimization is a trade off between minimizing errors betweenthe control variables and their desired values and minimizing thechanges in the independent variables. Without the move suppression term,the independent variable changes resulting from the set point changeswould be quite large due to the sudden and immediate error between thepredicted and desired values. Move suppression limits the independentvariable changes, but for all circumstances, not just the initialerrors.

The IdCom algorithm utilizes a different approach. Instead of a constantdesired value, a path is defined for the control variables to take fromthe current value to the desired value. This is illustrated in FIG. 16.This path is a more gradual transition from one operation point to thenext. Nevertheless, it is still a rigidly defined path that must be met.The objective function for this algorithm takes the form:$\begin{matrix}{\min\limits_{\Delta\quad U_{il}}{\sum\limits_{j}{\sum\limits_{k}\left( {A_{j}*\left( {Y^{P_{jk}} - y_{refjk}} \right)} \right)^{2}}}} & (25)\end{matrix}$

This technique is described in Richalet, J., A. Rault, J. L. Testud, andJ. Papon, Model Predictive Heuristic Control: Applications to IndustrialProcesses, Automatica, 14, 413-428 (1978), which is incorporated hereinby reference. It. should be noted that the requirement of Equation 25 ateach time interval is sometimes difficult. In fact, for controlvariables that behave similarly, this can result in quite erraticindependent variable changes due to the control algorithm attempting toendlessly meet the desired path exactly.

Control algorithms such as the DMC algorithm that utilize a form ofmatrix inversion in the control calculation, cannot handle controlvariable hard constraints directly. They must treat them separately,usually in the form of a steady-state linear program. Because this isdone as a steady-state problem, the constraints are time invariant bydefinition. Moreover, since the constraints are not part of a controlcalculation, there is no protection against the controller violating thehard constraints in the transient while satisfying them at steady-state.

With further reference to FIG. 15, the boundaries at the end of theenvelope can be defined as described hereinbelow. One techniquedescribed in the prior art, W. Edwards Deming, “Out of the Crisis,”Massachusetts Institute of Technology, Center for Advanced EngineeringStudy, Cambridge Mass., Fifth Printing, September 1988, pages 327-329,describes various Monte Carlo experiments that set forth the premisethat any control actions taken to correct for common process variationactually may have a negative impact, which action may work to increasevariability rather than the desired effect of reducing variation of thecontrolled processes. Given that any process has an inherent accuracy,there should be no basis to make a change based on a difference thatlies within the accuracy limits of the system utilized to control it. Atpresent, commercial controllers fail to recognize the fact that changesare undesirable, and continually adjust the process, treating alldeviation from target, no matter how small, as a special cause deservingof control actions, i.e., they respond to even minimal changes. Overadjustment of the manipulated variables therefore will result, andincrease undesirable process variation. By placing limits on the errorwith the present filtering algorithms described herein, only controlleractions that are proven to be necessary are allowed, and thus, theprocess can settle into a reduced variation free from unmeritedcontroller disturbances. The following discussion will deal with onetechnique for doing this, this being based on statistical parameters.

Filters can be created that prevent model-based controllers from takingany action in the case where the difference between the controlledvariable measurement and the desired target value are not significant.The significance level is defined by the accuracy of the model uponwhich the controller is statistically based. This accuracy is determinedas a function of the standard deviation of the error and a predeterminedconfidence level. The confidence level is based upon the accuracy of thetraining. Since most training sets for a neural network-based model willhave “holes” therein, this will result in inaccuracies within the mappedspace. Since a neural network is an empirical model, it is only asaccurate as the training data set. Even though the model may not havebeen trained upon a given set of inputs, it will extrapolate the outputand predict a value given a set of inputs, even though these inputs aremapped across a space that is questionable. In these areas, theconfidence level in the predicted output is relatively low. This isdescribed in detail in U.S. patent application Ser. No. 08/025,184,filed Mar. 2, 1993, which is incorporated herein by reference.

Referring now to FIG. 17, there is illustrated a flowchart depicting thestatistical method for generating the filter and defining the end point186 in FIG. 15. The flowchart is initiated at a start block 200 and thenproceeds to a function block 202, wherein the control values u(t+1) arecalculated. However, prior to acquiring these control values, thefiltering operation must be a processed. The program will flow to afunction block 204 to determine the accuracy of the controller. This isdone off-line by analyzing the model predicted values compared to theactual values, and calculating the standard deviation of the error inareas where the target is undisturbed. The model accuracy of e_(m)(t) isdefined as follows:e _(m)(t)=a(t)−p(t)   (26)where: e_(m)=model error,

a=actual value

p=model predicted value

The model accuracy is defined by the following equation:Acc=H*σ _(m)   (27)where: Acc=accuracy in terms of minimal detector error

H=significance level=1 67% confidence

-   -   =2 95% confidence    -   =3 99.5% confidence

σ_(m)=standard deviation of e_(m)(t).

The program then flows to a function block 206 to compare the controllererror e_(c)(t) with the model accuracy. This is done by taking thedifference between the predicted value (measured value) and the desiredvalue. This is the controller error calculation as follows:e _(c)(t)=d(t)−m(t)   (28)where: e_(c)=controller error

d=desired value

m=measured value

The program will then flow to a decision block 208 to determine if theerror is within the accuracy limits. The determination as to whether theerror is within the accuracy limits is done utilizing Shewhart limits.With this type of limit and this type of filter, a determination is madeas to whether the controller error e_(c)(t) meets the followingconditions: e_(c)(t)≧−1*Acc and e_(c)(t)≦+1*Acc, then either the controlaction is suppressed or not suppressed. If it is within the accuracylimits, then the control action is suppressed and the program flowsalong a “Y” path. If not, the program will flow along the “N” path tofunction block 210 to accept the u(t+1) values. If the error lies withinthe controller accuracy, then the program flows along the “Y” path fromdecision block 208 to a function block 212 to calculate the runningaccumulation of errors. This is formed utilizing a CUSUM approach. Thecontroller CUSUM calculations are done as follows:S _(low)=min(0, S _(low)(t−1)+d(t)−m(t))−Σ(m)+k)   (29)S _(hi)=max(0, S _(hi)(t−1)+[d(t)−m(t))−Σ(m)]−k)   (30)where: S_(hi)=Running Positive Qsum

S_(low)=Running Negative Qsum

k=Tuning factor—minimal detectable change threshold

with the following defined:

Hq=significance level. Values of (j,k) can be found so that the CUSUMcontrol chart will have significance levels equivalent to Shewhartcontrol charts.

The program will then flow to a decision block 214 to determine if theCUSUM limits check out, i.e., it will determine if the Qsum values arewithin the limits. If the Qsum, the accumulated sum error, is within theestablished limits, the program will then flow along the “Y” path. And,if it is not within the limits, it will flow along the “N” path toaccept the controller values u(t+1). The limits are determined if boththe value of S_(hi)≧+1*Hq and S_(low)≦1*Hq. Both of these actions willresult in this program flowing along the “Y” path. If it flows along the“N” path, the sum is set equal to zero and then the program flows to thefunction block 210. If the Qsum values are within the limits, it flowsalong the “Y” path to a function block 218 wherein a determination ismade as to whether the user wishes to perturb the process. If so, theprogram will flow along the “Y” path to the function block 210 to acceptthe control values u(t+1). If not, the program will flow along the “N”path from decision block 218 to a function block 222 to suppress thecontroller values u(t+1). The decision block 218, when it flows alongthe “Y” path, is a process that allows the user to re-identify the modelfor on-line adaptation, i.e., retrain the model. This is for the purposeof data collection and once the data has been collected, the system isthen reactivated.

Referring now to FIG. 18, there is illustrated a block diagram of theoverall optimization procedure. In the first step of the procedure, theinitial steady-state values {Y_(ss) ^(i), U_(ss) ^(i)} and the finalsteady-state values {Y_(ss) ^(f), U_(ss) ^(f)} are determined, asdefined in blocks 226 and 228, respectively. In some calculations, boththe initial and the final steady-state values are required. The initialsteady-state values are utilized to define the coefficients a^(i), b^(i)in a block 228. As described above, this utilizes the coefficientscaling of the b-coefficients. Similarly, the steady-state values inblock 228 are utilized to define the coefficients a^(f), b^(f), it beingnoted that only the b-coefficients are also defined in a block 229. Oncethe beginning and end points are defined, it is then necessary todetermine the path therebetween. This is provided by block 230 for pathoptimization. There are two methods for determining how the dynamiccontroller traverses this path. The first, as described above, is todefine the approximate dynamic gain over the path from the initial gainto the final gain. As noted above, this can incur some instabilities.The second method is to define the input values over the horizon fromthe initial value to the final value such that the desired value Y_(ss)^(f) is achieved. Thereafter, the gain can be set for the dynamic modelby scaling the b-coefficients. As noted above, this second method doesnot necessarily force the predicted value of the output y^(p)(t) along adefined path; rather, it defines the characteristics of the model as afunction of the error between the predicted and actual values over thehorizon from the initial value to the final or desired value. Thiseffectively defines the input values for each point on the trajectoryor, alternatively, the dynamic gain along the trajectory.

Referring now to FIG. 18 a, there is illustrated a diagrammaticrepresentation of the manner in which the path is mapped through theinput and output space. The steady-state model is operable to predictboth the output steady-state value Y_(ss) ^(i) at a value of k=0, theinitial steady-state value, and the output steady-state value Y_(ss)^(i) at a time t+N where k=N, the final steady-state value. At theinitial steady-state value, there is defined a region 227, which region227 comprises a surface in the output space in the proximity of theinitial steady-state value, which initial steady-state value also liesin the output space. This defines the range over which the dynamiccontroller can operate and the range over which it is valid. At thefinal steady-state value, if the gain were not changed, the dynamicmodel would not be valid. However, by utilizing the steady-state modelto calculate the steady-state gain at the final steady-state value andthen force the gain of the dynamic model to equal that of thesteady-state model, the dynamic model then becomes valid over a region229, proximate the final steady-state value. This is at a value of k=N.The problem that arises is how to define the path between the initialand final steady-state values. One possibility, as mentionedhereinabove, is to utilize the steady-state model to calculate thesteady-state gain at multiple points along the path between the initialsteady-state value and the final steady-state value and then define thedynamic gain at those points. This could be utilized in an optimizationroutine, which could require a large number of calculations. If thecomputational ability were there, this would provide a continuouscalculation for the dynamic gain along the path traversed between theinitial steady-state value and the final steady-state value utilizingthe steady-state gain. However, it is possible that the steady-statemodel is not valid in regions between the initial and final steady-statevalues, i.e., there is a low confidence level due to the fact that thetraining in those regions may not be adequate to define the modeltherein. Therefore, the dynamic gain is approximated in these regions,the primary goal being to have some adjustment of the dynamic modelalong the path between the initial and the final steady-state valuesduring the optimization procedure. This allows the dynamic operation ofthe model to be defined. This is represented by a number of surfaces 225as shown in phantom.

Referring now to FIG. 19, there is illustrated a flow chart depictingthe optimization algorithm. The program is initiated at a start block232 and then proceeds to a function block 234 to define the actual inputvalues u^(a)(t) at the beginning of the horizon, this typically beingthe steady-state value U_(ss). The program then flows to a functionblock 235 to generate the predicted values y^(p)(k) over the horizon forall k for the fixed input values. The program then flows to a functionblock 236 to generate the error E(k) over the horizon for all k for thepreviously generated y^(p)(k). These errors and the predicted values arethen accumulated, as noted by function block 238. The program then flowsto a function block 240 to optimize the value of u(t) for each value ofk in one embodiment. This will result in k-values for u(t). Of course,it is sufficient to utilize less calculations than the totalk-calculations over the horizon to provide for a more efficientalgorithm. The results of this optimization will provide the predictedchange Δu(t+k) for each value of k in a function block 242. The programthen flows to a function block 243 wherein the value of u(t+k) for eachu will be incremented by the value Δu(t+k). The program will then flowto a decision block 244 to determine if the objective function notedabove is less than or equal to a desired value. If not, the program willflow back along an “N” path to the input of function block 235 to againmake another pass. This operation was described above with respect toFIGS. 11 a and 11 b. When the objective function is in an acceptablelevel, the program will flow from decision block 244 along the “Y” pathto a function block 245 to set the value of u(t+k) for all u. Thisdefines the path. The program then flows to an End block 246.

Steady State Gain Determination

Referring now to FIG. 20, there is illustrated a plot of the input spaceand the error associated therewith. The input space is comprised of twovariables x₁ and x₂. The y-axis represents the function f(x₁, x₂). Inthe plane of x₁ and x₂, there is illustrated a region 250, whichrepresents the training data set. Areas outside of the region 250constitute regions of no data, i.e., a low confidence level region. Thefunction Y will have an error associated therewith. This is representedby a plane 252. However, the error in the plane 250 is only valid in aregion 254, which corresponds to the region 250. Areas outside of region254 on plane 252 have an unknown error associated therewith. As aresult, whenever the network is operated outside of the region 250 withthe error region 254, the confidence level in the network is low. Ofcourse, the confidence level will not abruptly change once outside ofthe known data regions but, rather, decreases as the distance from theknown data in the training set increases. This is represented in FIG. 21wherein the confidence is defined as α(x). It can be seen from FIG. 21that the confidence level α(x) is high in regions overlying the region250.

Once the system is operating outside of the training data regions, i.e.,in a low confidence region, the accuracy of the neural net is relativelylow. In accordance with one aspect of the preferred embodiment, a firstprinciples model g(x) is utilized to govern steady-state operation. Theswitching between the neural network model f(x) and the first principlemodels g(x) is not an abrupt switching but, rather, it is a mixture ofthe two.

The steady-state gain relationship is defined in Equation 7 and is setforth in a more simple manner as follows: $\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {f\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (31)\end{matrix}$

A new output function Y(u) is defined to take into account theconfidence factor α(u) as follows:Y({right arrow over (u)})=α({right arrow over (u)}).f({right arrow over(u)})+(1−α({right arrow over (u)}))g({right arrow over (u)})   (32)where: α(u)=confidence in model f(u)

α(u) in the range of 0→1

α(u) ε{0,1}

This will give rise to the relationship: $\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {Y\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (33)\end{matrix}$

In calculating the steady-state gain in accordance with this Equationutilizing the output relationship Y(u), the following will result:$\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = {{\frac{\partial\left( {\alpha\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)} \times {F\left( \overset{\rightarrow}{u} \right)}} + {{\alpha\left( \overset{\rightarrow}{u} \right)}\frac{\partial\left( {F\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} + {\frac{\partial\left( {1 - {\alpha\left( \overset{\rightarrow}{u} \right)}} \right)}{\partial\left( \overset{\rightarrow}{u} \right)} \times {g\left( \overset{\rightarrow}{u} \right)}} + {\left( {1 - {\alpha\left( \overset{\rightarrow}{u} \right)}} \right)\frac{\partial\left( {g\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}}}} & (34)\end{matrix}$

Referring now to FIG. 22, there is illustrated a block diagram of theembodiment for realizing the switching between the neural network modeland the first principles model. A neural network block 300 is providedfor the function f(u), a first principle block 302 is provided for thefunction g(u) and a confidence level block 304 for the function α(u).The input u(t) is input to each of the blocks 300-304. The output ofblock 304 is processed through a subtraction block 306 to generate thefunction 1−α(u), which is input to a multiplication block 308 formultiplication with the output of the first principles block 302. Thisprovides the function (1−α(u))*g(u). Additionally, the output of theconfidence block 304 is input to a multiplication block 310 formultiplication with the output of the neural network block 300. Thisprovides the function f(u)*α(u). The output of block 308 and the outputof block 310 are input to a summation block 312 to provide the outputY(u).

Referring now to FIG. 23, there is illustrated an alternate embodimentwhich utilizes discreet switching. The output of the first principlesblock 302 and the neural network block 300 are provided and are operableto receive the input x(t). The output of the network block 300 and firstprinciples block 302 are input to a switch 320, the switch 320 operableto select either the output of the first principals block 302 or theoutput of the neural network block 300. The output of the switch 320provides the output Y(u).

The switch 320 is controlled by a domain analyzer 322. The domainanalyzer 322 is operable to receive the input x(t) and determine whetherthe domain is one that is within a valid region of the network 300. Ifnot, the switch 320 is controlled to utilize the first principlesoperation in the first principles block 302. The domain analyzer 322utilizes the training database 326 to determine the regions in which thetraining data is valid for the network 300. Alternatively, the domainanalyzer 320 could utilize the confidence factor α(u) and compare thiswith a threshold, below which the first principles model 302 would beutilized.

Non-Linear Mill Control

Overall, model predictive control (MPC) has been the standardsupervisory control tool for such processes as are required in thecement industry. In the cement industry, particulate is fabricated witha kiln/cooler to generate raw material and then to grind this materialwith a mill. The overall kiln/cooler application, in the presentembodiment, utilizes a model of the process rather than a model of theoperator. This model will provide continuous regulation and disturbancerejection which will allow the application to recover from major upsets,such as coating drop three times faster than typical operatorintervention.

In general, mills demonstrate a severe non-linear behavior. This canpresent a problem in various aspects due to the fact that the gains atdifferent locations within the input space can change. The cement kilnsand coolers present a very difficult problem, in that the associatedprocesses, both chemical and physical, are in theory simple, but inpractice complex. This is especially so when commercial issues such asquality and costs of production are considered. The manufacturing ofcement, and its primary ingredient, clinker, has a number of conflictingcontrol objectives, which are to maximize production, minimize costs,and maximize efficiency, while at the same time maintaining minimumquality specifications. All of this optimization must take place withinvarious environmental, thermodynamic and mechanical constraints.

A primary technique of control for clinker has been the operator. Asrotary cement kilns and automation technology evolve, various automationsolutions have been developed for the cement industry. These solutionshave been successful to a greater or lessor extent. In the presentapplication, the process is modeled, rather than the operator, and modelpredictive control is utilized. Moves are made every control cycle tothe process based on continuous feedback of key measurements. This givesrise to a continuous MPC action, as opposed to the intermittent, albeitfrequent moves made by the typical expert system. In addition, as willbe described hereinbelow, the approach described utilizes fullmultivariable control (MVC) techniques, which take into account allcoupled interactions in the kiln/cooler process.

The cement mill is utilized to manufacture the various grades of cementafter processing of the raw material, which are defined by theirchemical composition and fineness (particle size distribution). Thecontrol objectives are thus to maximize production at minimum cost,i.e., low energy consumption for the various product grades, chemicalcompositions and specified fineness. In general, the mill utilizes aclosed circuit where separators in the feed-back are utilized toclassify the mill output into oversized and undersized product streams.The oversized stream, which does not conform to specification requiredfor correct cement strength, is fed back into the mill for furthergrinding and size reduction. Depending upon the type of mill, controlsinclude fresh feed, recirculating-load, as well as separator speed, allof which are used by the operator to control fineness, energyconsumption and throughput.

In general, the mill grinding equations take the form of:ln(P)=k ₁ +k ₂ *Fwhere:

P=particle size

F=feed rate

k₁ and k₂ are constants.

It has generally been stated in the literature that grinding modelequations are non-linear and hence, direct application of linear controltheory is not possible. The primary reason for this is that theoperation of the plant is only non-linear in very small regions of theinput space. Once the process has traversed, i.e., “stepped,” from oneportion of the input space to another portion thereof, the overall modelchanges, i.e., it is non-linear. This lends itself to non-linearmodeling techniques. However, most control systems are linear in nature,especially those that model the dynamics of the plant.

Referring now to FIG. 24, there is illustrated a diagrammatic view ofthe kiln/cooler configuration and the selected instrumentation utilizedfor optimal MPC control. This kiln/cooler consists of a five-stagesuspension pre-heater kiln, with back-end firing (approximately fifteenpercent of total firing). The cooler is a grate type with a conversionupgrade on the first section. It has on-line analyzers for NOx, O₂ andCO located at the top of a preheater 2402 which receives raw mealtherein. The output of the preheater is input to a kiln 2404. There isprovided a coal feed on an input 2406, the feed end and a coal feed 2408on the firing end. The output of the kiln is input to a cooler 2410which has an input cooler fan 2412. The output of the cooler providesthe clinker. The overall plan is fully instrumented with all necessarymeasurements, e.g., temperature, pressure and flow measurements such ascoal, raw meal and grate air. The quality of the clinker production ismaintained by the analysis of hourly samples of the raw meal feed,clinker and coal. This is supported by a semi-automated sampling system,and various modern laboratory infrastructure.

This system is controlled with an MPC controller (not shown) thatconsists of the MPC control engine as described hereinabove, as well asa real-time expert system that performs a variety of pre and postprocessing of control signals as well as various other functions such asMPC engine control, noise filtering, bias compensation and real-timetrending. This system will also perform set point tracking forbumperless transfer, and adaptive target selection. This allows for thecontroller tuning parameters to be changed according to various businessand/or process strategies.

The MPC is defined in two primary phases the first being the modelingphase in which the models of the kiln processes are developed. Thesecond phase is the deployment phase, where the models are commissionedand refined to a point where satisfactory control can be obtained.Central to commissioning is the tuning where the controller is tweakedto provide the desired control and optimization. For example this couldbe: maximum production at the expense of quality, or optimal quality atthe expense of efficiency.

The MPC models are developed from the analysis of test and process data,together with knowledge from the plant operators and other domainexperts. The result is a matrix of time responses, where each responsereflects the dynamic interaction of a controlled variable to amanipulated variable.

The tuning involves the selection of targets (setpoints), weightingfactors and various constraints for each variable. This determines howthe controller will solve the control problem at any given time. Thecontrol of the kiln and its optimization within the above set ofconstraints is solved every control cycle.

The solution chosen in a particular control cycle may not seem to benecessarily optimal at that given time, but will be optimal within thesolution space which has temporal as well as spatial dimensions. Thusthe control solution is a series of trajectories into the future, wherethe whole solution is optimized with time. The very nature of optimalcontrol in real time does not allow for a guarantee of a global optimalsolution. However the calculation of an optimal solution within a finiteamount of time is itself a class of optimization.

Some of the tuning parameters, which can be changed during operations,include:

-   -   1) Targets. Targets can be set for both controlled and        manipulated variables, and the MPC controller will try and force        all variables to their desired targets. In particular setting a        target for a manipulated variable such as coal allows for        optimization, and in particular efficiency, because the        controller will continually seek a lower coal flow while        maintaining production and quality. For some variables such as        O₂, a target may not be necessary, and the variables will be        allowed to roam within a band of constraints.    -   2) Priorities. Setting relative priorities between manipulated        variables and controlled variables allows the controller to        prioritize which are more important problems to solve, and what        type of solution to apply. Underspecified multivariable control        (more manipulated variables than controlled variables, as is the        case in this application) implies that for every problem there        will be more than one solution, but within constraints one        solution will generally be more optimal than others. For        example, too high a hood temperature can be controlled by, (a)        reducing fuel, (b) increasing the grate speed, or (c) increasing        cooler airflow, or a combination of the above.    -   3) Hard Constraints. Setting upper and lower hard constraints        for each process variable, for example, minimum and maximum        grate speed. These values which are usually defined by the        mechanical and operational limitations of the grate. Maintaining        these constraints is obviously realizable with controlled        variables such as ID-fan speed, but is more difficult to achieve        with, for example, hood temperature. However when hood        temperature exceeds a upper hard constraint of say 1200° C., the        controller will switch priority to this temperature excursion,        and all other control problems will “take a back seat” to the        solution required to bring this temperature back into the        allowable operating zone.    -   4) Soft upper and lower constraints. If any process variable        penetrates into the soft constraint area, penalties will be        incurred that will begin to prioritize the solution of this        problem. Continuous penetration into this area will cause        increasing prioritization of this problem, thus in effect        creating an adaptive prioritization, which changes with the        plant state.    -   5) Maximum rate of change constraints. These parameters are only        applicable to the manipulated variables, and generally reflect a        mechanical of physical limitation of the equipment used, for        example maximum coal feed rate.

From a clinker production point of view the functions of the MPCapplication can be viewed as follows:

-   -   1) Kiln Combustion Control where manipulated variables such as        ID-fan speed and fuel-flow are manipulated to control primarily        O₂. When CO rises above a specified threshold constraint, it        will override and become the controlled variable. The controller        is tuned to heavily penalize high CO values, steer the CO back        into an acceptable operating region, and rapidly return to O₂        control.    -   2) Kiln Thermal “Hinge Point” Control adjusts total coal, cooler        grate speed, and cooler fans to control the hood temperature.        The hood temperature is conceptualized as the “hinge” point on        which the kiln temperature profile hangs. The controller is        tuned to constantly minimize cooler grate speed and cooler fans,        so that heat recovery from the cooler is maximized, while        minimizing moves to coal.    -   3) Kiln Thermal “Swing Arm” Control adjusts percent coal to the        kiln backend, in order to control clinker free lime based on        hourly lab feedback. This control function is about three times        slower than the hinge point control, which maintains hood        temperature at a fixed target. The “swing arm effect” raises or        lowers the back end temperature with a constant firing end        temperature to compensate for changes in free lime. This is in        effect part of the quality control.

Kiln combustion control, kiln thermal hinge point control, and kilnthermal swing arm control are implemented in a single MPC controller.Kiln speed is included as a disturbance variable, as the productionphilosophy, in one embodiment, calls for setting a production rate tomeet various commercial obligations. This means that any changes to kilnspeed and hence production rate by the operator will be taken intoaccount in the MPC predictions, but the MPC controller will not be ableto move kiln speed.

The control system allows for customization of the interface between theplant and the MPC special control functions, the special controlfunctions implemented including:

-   -   1) Total Coal Control allows the operator to enter a total coal        or fuel flow setpoint. The control system “wrapper” splits the        move to the front and back individual coal flow controllers        while maintaining the percent of coal to the back constant. The        purpose of this control function is to allow heating and cooling        of the kiln while maintaining a constant energy profile from the        preheaters through to the firing end of the kiln. This provides        a solid basis for the temperature “hinge point” advanced control        function previously described.    -   2) Percent Coal to the Back Control allows the operator to enter        a percent coal to the back target and implements the moves to        the front and back coal flow controllers to enforce it. The        purpose of this control is to allow the back end temperature to        be swung up or down by the thermal “swing arm” advanced control        function.    -   3) Feed-to-Speed Ratio Control adjusts raw meal feed to the kiln        to maintain a constant ratio to kiln speed. The purpose of this        controller is to maintain a constant bed depth in the kiln,        which is important for long-term stabilization.    -   4) Cooler Fans Control is a move splitter that relates a single        generic cooler air fans setpoint to actual setpoints required by        n cooler air fans. The expert system wrapper through intelligent        diagnostics or by operator selection can determine which of the        air cooler fans will be placed under direct control of the MPC        controller, thus allowing for full control irrespective of the        (for example) maintenance being undertaken on any fans.    -   5) Gas analyzer selection. The control system automatically        scans the health of the gas analyzers, and will switch to the        alternative analyzer should the signals become “unhealthily’. In        addition the control system is used to intelligently extract the        fundamental control signals from the O₂ and CO readings, which        are badly distorted by purge spikes etc.

Referring now to FIG. 25, there is illustrated a block diagram of thenon-linear mill and the basic instrumentation utilized for advancedcontrol therein. The particle size overall is measured as “Blaine” incm²/gm, and is controlled by the operator through adjustment of thefresh feed rate and the separator speed. The mill is a ball mill, whichis referred to by reference numeral 2502. The fresh feed is metered by afresh feed input device 2506 which receives mined or processed materialinto the mill 2502, which mill is a mechanical device that grinds thismined or processed material. In this embodiment, the mill is a ballmill, which is a large cylinder that is loaded with steel balls. Themill 2502 rotates and is controlled by a motor 2508 and, as the materialpasses therethrough, it is comminuted to a specified fineness or Blaine.The output of the mill is input to an elevator 2510 which receives theoutput of the mill 2502 and inputs it to a separator 2512. This is afeedback system which is different than an “open circuit” mill and isreferred to as a “closed circuit” mill. The mill product is fed into theseparator 2512, which separator 2512 then divides the product into twostreams. The particles that meet product specification are allowed toexit the system as an output, represented by a reference numeral 2514,whereas the particles that are too large are fed back to the input ofthe mill 2502 through a return 2516 referred to as the course return.

There are provided various sensors for the operation of the mill. Theseparator speed is controlled by an input 2518 which signal is generatedby a controller 2520. The elevator 2510 provides an output 2522, whichconstitutes basically the current required by the elevator 2510. Thiscan be correlated to the output of the mill, as the larger the output,the more current that is required to lift it to the separator 2512.Additionally, the motor 2508 can provide an output. There isadditionally provided an “ear,” which is a sonic device that monitorsthe operation of the mill through various sonic techniques. It is knownthat the operation of the mill can be audibly detected such thatoperation within certain frequency ranges indicates that the mill isrunning well and in other frequency ranges that it is not running well,i.e., it is not optimum.

Overall, the mill-separator-return system is referred to as a “millcircuit.” The main control variable for a mill circuit is productparticle size, the output, and fresh feed is manipulated to control thisvariable. A secondary control variable is return and separator speed ismanipulated to control this variable. There are also provided variousconstants as inputs and constraints for the control operation. Thiscontroller 2520 will also control fresh feed on a line 2524.

The response of particle size to a move in fresh feed is known to beslow (one-two hours) and is dominated by dead time. Where a dead time totime constant ratio exceeding 0.5 is known to be difficult to controlwithout model predictive control techniques, documents ratios for theresponse of particle size to a move in fresh feed includes 0.9 and 1.3.

In the case of a closed-circuit mill, a move to fresh feed effects notonly the product particle size, but also the return flow. Also, a moveto separator speed effects not only the return flow, but also theparticle size. This is a fully interactive multi-variable controlproblem.

The controller adjusts fresh feed and separator speed to control Blaineand return. It also includes motor and sonic ear as outputs, and thesonic ear is currently used as a constraint variable. That means whenthe sonic ear decibel reading is too high then fresh feed is decreased.In this way the controller maximizes feed to the sonic ear (choking)constraint.

Referring now to FIG. 26, there is illustrated a dynamic model matrixfor the fresh feed and the separator speed for the measured variables ofthe Blaine Return Ear and motor. It can be seen that each of theseoutputs has a minimum and maximum gain associated therewith dead-timedelay and various time constants.

Referring now to FIG. 27, there is illustrated a plot of log sheet datautilized to generate a gain model for the controller 2520. Thisconstitutes the historical data utilized to train the steady statenon-linear model.

In general, the operation described hereinabove utilizes a non-linearcontroller which provides a model of the dynamics of the plants in aparticular region. The only difference in the non-linear model betweenone region of the input space to a second region of the input space isthat associated with the dynamic gain “k.” This dynamic gain varies asthe input space is traversed, i.e., the model is only valid over a smallregion of the input space for a given dynamic gain. In order tocompensate for this dynamic gain of a dynamic linear model, i.e., thecontroller, a non-linear steady state model of the overall process isutilized to calculate a steady-state gain “K” which is then utilized tomodify the dynamic gain “k.” This was described in detail hereinabove.In order to utilize this model, it is necessary to first model thenon-linear operation of the process, i.e., determining a non-linearsteady state model, and then also determine the various dynamics of thesystem through “step testing.” The historical data provided by the logsheets of FIG. 27 provide this information which can then be utilized totrain the non-linear steady state model.

Referring now to FIGS. 28 a and 28 b, there are illustrated plots ofprocess gains for the overall non-linear mill model with the scalesaltered. In FIG. 28 a, there is illustrated a plot of the sensitivityversus the percent move of the Blaine. There are provided two plots, onefor separator speed and one for fresh feed. The upper plot illustratesthat the sensitivity of Blaine to separator speed is very low, whereasthe gain of the percent movement of fresh feed with respect to theBlaine varies considerably. It can be seen that at a minimum, the gainis approximately −1.0 and then approaches 0.0 as Blaine increases. Inthe plot of FIG. 28 b, there are illustrated two plots for thesensitivity versus the percent return. In the upper plot, thatassociated with the percent movement of separator speed, it can be seenthat the gain varies from a maximum of 0.4 at a low return to a gain of0.0 at a high return value. The fresh feed varies similarly, with lessrange. In general, the plots of FIGS. 28 a and 28 a, it can be seen thatthe sensitivity of the control variable, e.g., Blaine in FIG. 28 a andreturn in the plot of FIG. 28 b, as compared to the manipulatedvariables of separator speed and fresh feed. The manipulated variablesare illustrated as ranging from their minimum to maximum values.

In operation of the controller 2520, the process gains (sensitivities)are calculated from the neural network model (steady state model) ateach control execution, and downloaded into the predictive models in thelinear controller (dynamic model). This essentially provides the linearcontroller with the dynamic gains that follow the sensitivitiesexhibited by the steady-state behavior of the process, and therebyprovides non-linear mill control utilizing a linear controller. Withsuch a technique, the system can now operate over a much broaderspecification without changing the tuning parameters or the predictivemodels in the controller.

Referring now to FIG. 29, there is illustrated a plot of non-lineargains for fresh feed responses for both manipulatible variables andcontrol variables. The manipulatible variable for the fresh feed isstepped from a value of 120.0 to approximately 70.0. It can be seen thatthe corresponding Blaine output and the return output, the controlledvariables, also steps accordingly. However, it can be seen that a stepsize of 10.0 in the fresh feed input does not result in identical stepin either the Blaine or the return over the subsequent steps. Therefore,it can be seen that this is clearly a non-linear system with varyinggains.

Referring now to FIG. 30, there is illustrated four plots of thenon-linear gains for separator speed responses, wherein the separatorspeed as a manipulatible variable is stepped from a value ofapproximately 45.0 to 85.0, in steps of approximately 10.0. It can beseen that the return controlled variable makes an initial change with aninitial step that is fairly large compared to any change at the end. TheBlaine, by comparison, appears to change an identical amount each time.However, it can be seen that the response of the return with respect tochanges in the separator speed will result in a very non-linear system.

Referring now to FIG. 31, there is illustrated an overall output for theclosed loop control operation wherein a target change for the Blaine isprovided. In this system, it can be seen that the fresh feed is steppedat a time 3102 with a defined trajectory and this results in the Blainefalling and the return rising and falling. Further, the separator speedis also changed from one value to a lower value. However, themanipulatible variables have the trajectory thereof defined by thenon-linear controller such that there is an initial point and a finalpoint and a trajectory therebetween. By correctly defining thistrajectory between the initial point and the final point, the trajectoryof the Blaine and the return can be predicted. This is due to the factthat the non-linear model models the dynamics of the plant, thesedynamics being learned from the step tests on the plant. This, ofcourse, is learned at one location in the input space and, by utilizingthe steady state gains from the non-linear steady state model tomanipulate the dynamic gains of the linear model, control can beeffected over different areas of the input space, even though theoperation is non-linear from one point in the input space to anotherspace.

Although the preferred embodiment has been described in detail, itshould be understood that various changes, substitutions and alterationscan be made therein without departing from the spirit and scope of theinvention as defined by the appended claims.

Further Embodiments

For nonlinear systems, process gain may be a determining characteristicof the systems, and may vary significantly over their respectiveoperating regions. However, in systems where the process gain has littlevariance, the system may be represented by a linearization of theprocess model. In various embodiments, the linearization may be localand/or global. For example, a global linearization may be desirable insystems where the process gains do not vary significantly over theentire operation range. Alternatively, for systems that aresignificantly nonlinear over the entire operation range, a locallinearization may be used about a specific operating point, e.g., at afirst time, such as the current operating point or some other desiredoperating point. Note that the accuracy of the linearization may dependon the nonlinearity of the system and may be defined based on how muchthe operating conditions differ from those represented in the model,i.e., in the linearization. Since the linear approximation does containsome error, it is desirable from a process control standpoint to choosea well-understood linearization. For example, in one case, it may bemore desirable to have the error be small at the current operating point(at the first time) at the expense of larger error at future operatingpoints. In another case it may be desirable to have the error be smallat the desired final operating point, e.g., at a second time, at theexpense of a larger error at the current operating point. Alternatively,it may be desirable to distribute the error so that it is more or lessequal across the entire operating region. In general, the linearizationmay be any function of the gain at the current point, the desired finalpoint, and the intermediate operating points.

For example, in one embodiment, the linearization may be based on asimple or weighted average between gains at two or more operatingpoints, e.g., between the gain at the current point and the final point.In some embodiments, gains may be approximated by a linear interpolationbetween a first gain, e.g., at the current operating point, and a latergain, e.g., at the final destination point. Note that in someembodiments the linearization may be based on a single operating point,or, alternatively, a plurality of operating points, in the operatingregion.

In some embodiments, it may be desirable for model based processcontrollers to utilize process gains that are larger in magnitude thanthe actual process gain, i.e., of the actual plant or process. Morespecifically, because the size of a control move, e.g., move(s) inresponse to determined offset(s) from target operation value(s), may bebased on a multiplication of error and the inverse of the process gain,larger process gain values may be implemented in order to restrict thesize of the control move. Thus, a larger process gain will result insmaller control moves, which, in turn, cause smaller adjustments anddisruptions to the operation of the plant or process. Thus, in someembodiments, the gain may be based on the largest absolute value of thegains at one or more operating points, such as the first and the lastgain in the change of operation.

Note that the approximation methods described above regarding theprocess gains are not limited to simple linear approximations, but, infact, maybe approximated via nonlinear methods as well, such as, forexample, a nonlinear interpolation between two gains in the operatingregion, e.g., the gain at the current operating point and the desiredfinal operating point.

Thus, the system may be modeled using various approximation methods,e.g., based on global and/or local linear approximations of dynamicgains based on gains at two operating points.

In some embodiments, the ratio of the gain at the current point and thedesired final operating point, e.g., at first and second times,respectively, may be used to measure the nonlinearity of the system.Because a more nonlinear system will, in general, be subject to moremodeling error, it may be desirable to configure or control otherparameters of the process control system based on the degree ofnonlinearity. In some embodiments, as indicated above, the error may beattenuated as a function of time, and/or of gain values associatedtherewith, between the current operating point and the desired finaloperating point. Alternatively, or additionally, the amount ofattenuation may be modified according to the nonlinearity of the system,e.g., a high attenuation may be used when the model is more nonlinearand correspondingly more prone to modeling error. In some embodiments, ahigher attenuation may also result in small control moves as theimportance of the control error may be reduced. Conversely, lowattenuation may be used when the model is more linear and so less proneto modeling error. In various embodiments, the attenuation may bemodified or adjusted according to various methods. For example, similarto above, the attenuation may be based on the values of gain at a firstoperating point, e.g., at the first time, and a later operating point,e.g., at the second time. Also similar to the above descriptionsregarding model approximations, the attenuation may be based on a simpleor weighted average of, a largest absolute value of, a linear ornonlinear interpolation between, and/or simply an operational pointvalue of, one or more properties of values between the current operatingpoint and the desired final operating point inclusively, e.g., the gainsof these values, among others.

Thus, the error may be attenuated according to various methods tofacilitate smoother transitions between operating points of a plant orprocess.

In some embodiments, the system may have an inherent accuracy, e.g., thedifference between the behavior of the actual system (plant or process)and the model of that system, which may be determined by comparison ofvalues of the model with actual process values. If the system ischanging dramatically in a nonlinear manner, and the corresponding modelis also changing in a nonlinear manner, it can generally be expectedthat the accuracy will be lower than if the system and model arechanging linearly.

Correspondingly, in some embodiments, the system may include an errortolerance level. For example, if the system is within a specifiedtolerance level of the desired operating point as measured from theplant or process and/or as predicted the model, no further controladjustments may be implemented, e.g., via a filter in the errorminimization device described above, because the model may not beaccurate enough to resolve small differences in the system. For a highlynonlinear system, this tolerance level may vary depending on the degreeof nonlinearity. For example, as the nonlinearity increases, the modelaccuracy may decrease, and a larger tolerance may be justified. Thus,the accuracy of the model, as determined from the gain values, maydetermine whether or not the error minimization device is used oremployed, e.g., may act as a filter regarding operation of the errorminimization device.

In one embodiment, the degree of nonlinearity and/or the tolerance maybe based on the dynamic gains at a first and second time, e.g., as afunction of the gains at these respective times. In some embodiments,the degree of nonlinearity and/or tolerance may based on a function ofthe dynamic gain k as the system moves from one operating point toanother operating point, e.g., the ratio of the gains, and/or magnitudeof the gains, among others, at the first and second times. Note that thevarious approximation and error attenuation methods described above mayalso be applied to the determination of the degree of nonlinearityand/or tolerance, e.g., via simple and weighted averages, largestabsolute values, linear and/or nonlinear interpolations, one or moreoperational point values, among others, e.g., of or between the gains atthe first and second operating point values inclusively.

Note further that each of these methods may also be applied indetermining an error constraint, e.g., an error frustum, for the model,e.g., such as those already described herein. More specifically, theerror constraint may be used for imposing a constraint on the objectivefunction, e.g., a hard and/or soft constraint. Thus, when the error isgreater than the error constraint, the constraint may be “activated” inthe objective function, i.e., a scalar factor may be applied to theconstraint as a weight for error minimization. Similar to above, theerror constraint may be based on operational point values, or propertiesthereof, at various points in time, e.g., such as the first and secondtimes described above. Thus, similar to above, the error constraint maybe based on the gains at the first and second times, e.g., via simpleand weighted averages, ratios, e.g., of the first and second respectivegains, largest absolute values, linear and/or nonlinear interpolations,one or more specific gain values, one or more magnitudes of the gains,etc.

Thus, the error of the model may be minimized via various constraints,filters, and/or other methods.

1. A method for controlling a nonlinear system, comprising: a linearcontroller receiving inputs representing measured variables of thesystem, wherein values of the measured variables of the system are in afirst operating region, wherein the linear controller is operable todetermine predicted control values for manipulable variables thatcontrol the system, and wherein the linear controller has a linear gaink; executing a nonlinear steady-state model of the system, wherein thenonlinear steady-state model has been trained to represent operation ofthe system over a specified region of the system's operating space, andwherein the nonlinear steady state model has a steady-state gain K;initiating a change in operation of the system, wherein the changeoperates to move the values of the measured variables of the system froma first operating point to a second operating point, wherein saidinitiating comprises modifying the nonlinear model in accordance withthe change in operation of the system; determining values of thesteady-state gain K at the first and second operating points; adjustingthe linear gain k of the linear controller as a function of the valuesof the steady-state gain K of the nonlinear steady-state model at thefirst and second operating points; the linear controller determiningpredicted control values for the manipulable variables in response tosaid adjusting; and outputting the predicted control values for themanipulable variables, wherein the predicted control values are useableto control the system.
 2. The method of claim 1, wherein the linear gaink of the linear controller is adjusted according to a simple average ofthe values of the steady-state gain K of the nonlinear steady-statemodel at the first and second operating points.
 3. The method of claim1, wherein the linear gain k of the linear controller is adjustedaccording to a larger absolute value of the values of the steady-stategain K of the nonlinear steady-state model at the first and secondoperating points.
 4. The method of claim 1, wherein the linear gain k ofthe linear controller is adjusted according to a weighted average of thevalues of the steady-state gain K of the nonlinear steady-state model atthe first and second operating points.
 5. The method of claim 1, whereinthe linear gain k of the linear controller is adjusted according to oneor more values between the values of the steady-state gain K of thenonlinear steady-state model at the first and second operating pointsinclusively.
 6. The method of claim 1, wherein the linear gain k of thelinear controller is adjusted according to a linear interpolationbetween the values of the steady-state gain K of the nonlinearsteady-state model at the first and second operating points inclusively.7. The method of claim 1, wherein the linear gain k of the linearcontroller is adjusted according to a nonlinear interpolation betweenthe values of the steady-state gain K of the nonlinear steady-statemodel at the first and second operating points inclusively.
 8. Themethod of claim 1, the method further comprising: modifying one or moreadditional parameters of the linear controller as a function of thevalues of the steady-state gain K of the nonlinear steady-state model atthe first and second operating points, wherein said linear controllerdetermining predicted control variables is further based on saidmodifying.
 9. The method of claim 1, further comprising: providing thepredicted control values as input to the system to control the system.10. The method of claim 1, further comprising: adjusting one or moreadditional parameters of the linear controller based on an associatedone or more parameters of the nonlinear model as the values of themeasured variables move.
 11. The method of claim 10, wherein saidadjusting one or more parameters of the linear controller based on anassociated one or more parameters of the nonlinear model as the valuesof the measured variables move comprises: adjusting allowed changes inthe predicted control values for the manipulable variables.
 12. Themethod of claim 1, wherein the change in operation of the systemcomprises one or more of: startup of the system; and changing anobjective of the system.
 13. The method of claim 12, wherein saidchanging an objective of the system comprises one or more of: changing aproduct; and changing a product grade.
 14. The method of claim 1,further comprising: controlling the system using the predicted controlvalues for the manipulable variables.
 15. The method of claim 14,further comprising: performing said receiving, said executing, saidinitiating, said adjusting, said determining, said outputting, and saidcontrolling in an iterative manner to operate the system.
 16. The methodof claim 1, wherein the linear controller is operable to model thedynamics of the system.
 17. The method of claim 1, wherein the specifiedregion of the system's operating space over which the nonlinear modelhas been trained to represent the system is greater than that over whichthe linear controller is valid.
 18. A system for controlling a nonlinearsystem, comprising: means for a linear controller receiving inputsrepresenting measured variables of a nonlinear system, wherein values ofthe measured variables of the nonlinear system are in a first operatingregion, wherein the linear controller is operable to determine predictedcontrol values for manipulable variables that control the nonlinearsystem, and wherein the linear controller has a linear gain k; means forexecuting a nonlinear steady-state model of the nonlinear system,wherein the nonlinear steady-state model has been trained to representoperation of the nonlinear system over a specified region of thenonlinear system's operating space, and wherein the nonlinear steadystate model has a steady-state gain K; means for initiating a change inoperation of the nonlinear system, wherein the change operates to movethe values of the measured variables of the nonlinear system from afirst operating point to a second operating point, wherein saidinitiating comprises modifying the nonlinear model in accordance withthe change in operation of the nonlinear system; means for determiningvalues of the steady-state gain K at the first and second operatingpoints; means for adjusting the linear gain k of the linear controlleras a function of the values of the steady-state gain K of the nonlinearsteady-state model at the first and second operating points; means forthe linear controller determining predicted control values for themanipulable variables in response to said adjusting; and means foroutputting the predicted control values for the manipulable variables,wherein the predicted control values are useable to control thenonlinear system.
 19. A system for operating a nonlinear system, thesystem comprising: a linear controller, operable to: receive inputsrepresenting measured variables of a nonlinear system, wherein values ofthe measured variables of the nonlinear system are in a first operatingregion; and determine predicted control values for manipulable variablesthat control the nonlinear system, and wherein the linear controller hasa linear gain k; and a nonlinear steady-state model of the nonlinearsystem, coupled to the linear controller, wherein the nonlinearsteady-state model has been trained to represent operation of thenonlinear system over a specified region of the nonlinear system'soperating space, and wherein the nonlinear steady state model has asteady-state gain K; wherein, in response to an initiated change inoperation of the nonlinear system from a first operating point to asecond operating point, including a modification to the nonlinear modelin accordance with the change in operation of the nonlinear system, thelinear controller is further operable to: adjust the linear gain k as afunction of values of the steady-state gain K of the nonlinearsteady-state model at first and second operating points; outputpredicted control values for the manipulable variables based on theadjusted linear gain k, wherein the predicted control values are useableto control the nonlinear system.
 20. A computer-accessible memory mediumthat stores program instructions for operating a nonlinear system,wherein the program instructions implement: a linear controller,operable to: receive inputs representing measured variables of anonlinear system, wherein values of the measured variables of thenonlinear system are in a first operating region; and determinepredicted control values for manipulable variables that control thenonlinear system, and wherein the linear controller has a linear gain k;and a nonlinear steady-state model of the nonlinear system, wherein thenonlinear steady-state model has been trained to represent operation ofthe nonlinear system over a specified region of the nonlinear system'soperating space, and wherein the nonlinear steady state model has asteady-state gain K, wherein the program instructions are executable bya processor to perform: executing the nonlinear steady-state model ofthe system; initiating a change in operation of the system, wherein thechange operates to move the values of the measured variables of thesystem from a first operating point to a second operating point, whereinsaid initiating comprises modifying the nonlinear model in accordancewith the change in operation of the system; determining values of thesteady-state gain K at the first and second operating points; adjustingthe linear gain k of the linear controller as a function of the valuesof the steady-state gain K of the nonlinear steady-state model at thefirst and second operating points; the linear controller determiningpredicted control values for the manipulable variables in response tosaid adjusting; and outputting the predicted control values for themanipulable variables, wherein the predicted control values are useableto control the system.